Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.
By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.
\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.
\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).
\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.
The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.
\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.
Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv
Source: https://github.com/pcm-dpc/COVID-19
url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19IT <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19IT$data <- as.Date(COVID19IT$data)
DT::datatable(COVID19IT)Warnings
- 29/03/2020: dati Regione Emilia Romagna parziali (dato tampone non aggiornato).
- 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
- 18/03/2020: dati Regione Campania non pervenuti.
- 18/03/2020: dati Provincia di Parma non pervenuti.
- 17/03/2020: dati Provincia di Rimini non aggiornati
- 16/03/2020: dati P.A. Trento e Puglia non pervenuti.
- 11/03/2020: dati Regione Abruzzo non pervenuti.
- 10/03/2020: dati Regione Lombardia parziali.
- 07/03/2020: dati Brescia +300 esiti positivi
# create data for analysis
data = data.frame(date = COVID19IT$data,
y = COVID19IT$totale_casi)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 3423.884423 445.939592 7.678 0.00000000521 ***
## th2 0.096123 0.003982 24.141 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5806 on 35 degrees of freedom
##
## Number of iterations to convergence: 8
## Achieved convergence tolerance: 0.000007606mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 130721.67709 1573.49293 83.08 <2e-16 ***
## xmid 29.22506 0.16006 182.58 <2e-16 ***
## scal 5.46071 0.06844 79.78 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 688.6 on 34 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000002714mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 1000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 225389.531906 10308.698272 21.86 <2e-16 ***
## b2 9.230192 0.285903 32.28 <2e-16 ***
## b3 0.933940 0.002173 429.73 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 946.3 on 34 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000004716richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "plinear",
control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging...
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 289001.527372 30398.371511 9.507 4.19e-11 ***
## th2 0.048984 0.004308 11.370 3.95e-13 ***
## th3 5.505691 0.405070 13.592 2.65e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1204 on 34 degrees of freedom
##
## Number of iterations to convergence: 6
## Achieved convergence tolerance: 0.005745
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3,
# data = data, start = start, trace = TRUE)models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -372.1411 | 3 | 0.9774689 | 750.2822 | 751.0095 | 755.1150 | |
| Logistic model | -292.7167 | 4 | 0.9996783 | 593.4334 | 594.6834 | 599.8770 | *** |
| Gompertz model | -304.4797 | 4 | 0.9993675 | 616.9593 | 618.2093 | 623.4030 | |
| Richards model | -313.3908 | 4 | 0.9990398 | 634.7816 | 636.0316 | 641.2253 |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Infected", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 5000),
minor_breaks = seq(0, coef(mod2)[1], by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top")last_plot() +
scale_y_continuous(trans = "log10", limits = c(100,NA)) +
labs(y = "Infected (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
minor_breaks = seq(0, max(ylim), by = 5000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 38 2020-04-01 132083 115834 150292
## 381 2020-04-01 108889 106785 110583
## 382 2020-04-01 113323 110946 115994
## 383 2020-04-01 114002 110720 117751
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Infected", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19IT$data,
y = COVID19IT$deceduti)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 187.252995 24.999428 7.49 0.000000009 ***
## th2 0.115859 0.003992 29.02 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 511.1 on 35 degrees of freedom
##
## Number of iterations to convergence: 10
## Achieved convergence tolerance: 0.000004745mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 16848.99386 401.24060 41.99 <2e-16 ***
## xmid 31.99358 0.27397 116.78 <2e-16 ***
## scal 5.11536 0.09699 52.74 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 107.1 on 34 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000009135mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
# control = nls.control(maxiter = 10000))
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 37372.976382 1645.111061 22.72 <2e-16 ***
## b2 11.494777 0.251808 45.65 <2e-16 ***
## b3 0.938466 0.001498 626.62 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 62.46 on 34 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.0000003555richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 53763.206073 5346.394867 10.06 0.0000000000101 ***
## th2 0.044279 0.002804 15.79 < 2e-16 ***
## th3 6.744370 0.337508 19.98 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 75.11 on 34 degrees of freedom
##
## Number of iterations to convergence: 3
## Achieved convergence tolerance: 0.000001336models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -282.2224 | 3 | 0.9864516 | 570.4449 | 571.1721 | 575.2776 | |
| Logistic model | -223.8678 | 4 | 0.9993952 | 455.7356 | 456.9856 | 462.1793 | |
| Gompertz model | -203.9114 | 4 | 0.9997592 | 415.8228 | 417.0728 | 422.2665 | *** |
| Richards model | -210.7370 | 4 | 0.9996690 | 429.4741 | 430.7241 | 435.9177 |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Deceased", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 500),
minor_breaks = seq(0, coef(mod2)[1], by = 100)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top")last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Deceased (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 38 2020-04-01 15292 13805 17024
## 381 2020-04-01 12871 12499 13133
## 382 2020-04-01 13356 13160 13558
## 383 2020-04-01 13431 13186 13683
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Deceased", color = "Model") +
scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# create data for analysis
data = data.frame(date = COVID19IT$data,
y = COVID19IT$dimessi_guariti)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
##
## Formula: y ~ exponential(x, th1, th2)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 245.518117 28.969849 8.475 0.000000000533 ***
## th2 0.114341 0.003533 32.364 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 571.1 on 35 degrees of freedom
##
## Number of iterations to convergence: 10
## Achieved convergence tolerance: 0.000002655mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
##
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 22720.6296 932.1895 24.37 <2e-16 ***
## xmid 32.9435 0.4845 68.00 <2e-16 ***
## scal 5.5073 0.1526 36.09 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 192.7 on 34 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000000281mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
##
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## Asym 61684.444346 7155.762297 8.62 0.000000000452 ***
## b2 10.062461 0.334618 30.07 < 2e-16 ***
## b3 0.947579 0.002876 329.44 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 157.8 on 34 degrees of freedom
##
## Number of iterations to convergence: 0
## Achieved convergence tolerance: 0.000001423richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2)
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss,
y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
# trace = TRUE, # algorithm = "port",
control = nls.control(maxiter = 1000))
summary(mod4)
##
## Formula: y ~ richards(x, th1, th2, th3)
##
## Parameters:
## Estimate Std. Error t value Pr(>|t|)
## th1 129801.652737 43117.557559 3.010 0.00489 **
## th2 0.029354 0.005309 5.529 0.000003530088115 ***
## th3 5.131000 0.457341 11.219 0.000000000000566 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 170.8 on 34 degrees of freedom
##
## Number of iterations to convergence: 22
## Achieved convergence tolerance: 0.000002203models = list("Exponential model" = mod1,
"Logistic model" = mod2,
"Gompertz model" = mod3,
"Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
df = sapply(models, function(m) attr(logLik(m), "df")),
Rsquare = sapply(models, function(m)
cor(data$y, fitted(m))^2),
AIC = sapply(models, AIC),
AICc = sapply(models, AICc),
BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)| loglik | df | Rsquare | AIC | AICc | BIC | ||
|---|---|---|---|---|---|---|---|
| Exponential model | -286.3357 | 3 | 0.9887711 | 578.6715 | 579.3987 | 583.5042 | |
| Logistic model | -245.6010 | 4 | 0.9985436 | 499.2020 | 500.4520 | 505.6456 | |
| Gompertz model | -238.2139 | 4 | 0.9989825 | 484.4278 | 485.6778 | 490.8715 | *** |
| Richards model | -241.1444 | 4 | 0.9988523 | 490.2888 | 491.5388 | 496.7325 |
ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(aes(y = fitted(mod1), color = "Exponential")) +
geom_line(aes(y = fitted(mod2), color = "Logistic")) +
geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
geom_line(aes(y = fitted(mod4), color = "Richards")) +
labs(x = "", y = "Recovered", color = "Model") +
scale_color_manual(values = cols) +
scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 500),
minor_breaks = seq(0, coef(mod2)[1], by = 100)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top")last_plot() +
scale_y_continuous(trans = "log10", limits = c(10,NA)) +
labs(y = "Recovered (log10 scale)")df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
fit1 = predict(mod1, newdata = df),
fit2 = predict(mod2, newdata = df),
fit3 = predict(mod3, newdata = df),
fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
coord_cartesian(ylim = ylim) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))
pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]
pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]
pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]
pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]
# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
subset(pred2, x == max(data$x)+1, select = 2:5),
subset(pred3, x == max(data$x)+1, select = 2:5),
subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
## date fit lwr upr
## 38 2020-04-01 18927 17166 20766
## 381 2020-04-01 16238 15670 16784
## 382 2020-04-01 16804 16393 17268
## 383 2020-04-01 16916 16455 17462
ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))ggplot(data, aes(x = date, y = y)) +
geom_point() +
geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
coord_cartesian(ylim = c(0, max(ylim))) +
labs(x = "", y = "Recovered", color = "Model") +
scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
minor_breaks = seq(0, max(ylim), by = 1000)) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = cols) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))df = data.frame(date = COVID19IT$data,
swabs = c(NA, diff(COVID19IT$tamponi)),
positives = c(NA, diff(COVID19IT$totale_casi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
df$r = df$positives/df$swabs
df = subset(df, swabs > 50)
graph1 <- ggplot(df, aes(x = date)) +
geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
geom_line(aes(y = swabs, color = "swabs")) +
geom_point(aes(y = positives, color = "positives"), pch = 15) +
geom_line(aes(y = positives, color = "positives")) +
labs(x = "", y = "Number of cases", color = "") +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
scale_color_manual(values = palette()[c(2,1)]) +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))
graph2 <- ggplot(df, aes(x = date, y = r)) +
geom_smooth(method = "loess", se = TRUE, col = "darkgrey") +
geom_point(col=palette()[4]) +
geom_line(size = 0.5, col=palette()[4]) +
labs(x = "", y = "New positives / swabs") +
scale_y_continuous(labels = scales::percent_format()) +
scale_x_date(date_breaks = "2 day", date_labels = "%b%d",
minor_breaks = "1 day") +
theme_bw() +
theme(legend.position = "top",
axis.text.x = element_text(angle=60, hjust=1))
grid.arrange(graph1, graph2, nrow = 2, ncol = 1, widths = 1, heights = c(0.6,0.4))